# Properties

 Label 3.331.4t5.2c1 Dimension 3 Group $S_4$ Conductor $331$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $331$ Artin number field: Splitting field of $f= x^{6} - x^{5} - 2 x^{3} - x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Odd Determinant: 1.331.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{2} + 12 x + 2$
Roots:
 $r_{ 1 }$ $=$ $2 + 10\cdot 13 + 2\cdot 13^{2} + 13^{3} + 5\cdot 13^{4} + 4\cdot 13^{6} + 2\cdot 13^{7} + 6\cdot 13^{8} +O\left(13^{ 9 }\right)$ $r_{ 2 }$ $=$ $6 + 2\cdot 13 + 13^{2} + 8\cdot 13^{3} + 8\cdot 13^{5} + 11\cdot 13^{6} + 12\cdot 13^{7} + 5\cdot 13^{8} +O\left(13^{ 9 }\right)$ $r_{ 3 }$ $=$ $a + 2 + \left(3 a + 9\right)\cdot 13 + \left(11 a + 8\right)\cdot 13^{2} + \left(10 a + 8\right)\cdot 13^{3} + \left(10 a + 8\right)\cdot 13^{4} + \left(6 a + 2\right)\cdot 13^{5} + \left(9 a + 8\right)\cdot 13^{6} + \left(8 a + 9\right)\cdot 13^{7} + \left(10 a + 5\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$ $r_{ 4 }$ $=$ $12 a + 3 + \left(9 a + 11\right)\cdot 13 + \left(a + 3\right)\cdot 13^{2} + \left(2 a + 8\right)\cdot 13^{3} + \left(2 a + 8\right)\cdot 13^{4} + \left(6 a + 11\right)\cdot 13^{5} + \left(3 a + 10\right)\cdot 13^{6} + \left(4 a + 8\right)\cdot 13^{7} + \left(2 a + 7\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$ $r_{ 5 }$ $=$ $8 a + 3 + \left(3 a + 5\right)\cdot 13 + \left(4 a + 4\right)\cdot 13^{2} + \left(9 a + 10\right)\cdot 13^{3} + \left(8 a + 1\right)\cdot 13^{4} + \left(10 a + 7\right)\cdot 13^{5} + \left(12 a + 7\right)\cdot 13^{6} + \left(6 a + 5\right)\cdot 13^{7} + \left(4 a + 1\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$ $r_{ 6 }$ $=$ $5 a + 11 + 9 a\cdot 13 + \left(8 a + 5\right)\cdot 13^{2} + \left(3 a + 2\right)\cdot 13^{3} + \left(4 a + 1\right)\cdot 13^{4} + \left(2 a + 9\right)\cdot 13^{5} + 9\cdot 13^{6} + \left(6 a + 12\right)\cdot 13^{7} + \left(8 a + 11\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,6,4)(2,3,5)$ $(1,3,4)(2,6,5)$ $(3,5)(4,6)$ $(3,4)(5,6)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $3$ $3$ $2$ $(1,2)(3,6)$ $-1$ $6$ $2$ $(3,5)(4,6)$ $1$ $8$ $3$ $(1,6,4)(2,3,5)$ $0$ $6$ $4$ $(1,3,2,6)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.